Abstract

For positive-sense single-stranded RNA virus genomes, there is a trade-off between the mutually exclusive tasks of transcription, translation and encapsidation. The replication strategy that maximizes the intracellular growth rate of the virus requires iterative genome transcription from positive to negative, and back to positive sense. However, RNA viruses experience high mutation rates, and the proportion of genomes with lethal mutations increases with the number of replication cycles. Thus, intracellular mutant frequency will depend on the replication strategy. Introducing apparently realistic mutation rates into a model of viral replication demonstrates that strategies that maximize viral growth rate could result in an average of 26 mutations per genome by the time plausible numbers of positive strands have been generated, and that virus viability could be as low as 0.1 per cent. At high mutation rates or when a high proportion of mutations are deleterious, the optimal strategy shifts towards synthesizing more negative strands per positive strand, and in extremis towards a ‘stamping-machine’ replication mode where all the encapsidated genomes come from only two transcriptional steps. We conclude that if viral mutation rates are as high as current estimates suggest, either mutation frequency must be considerably higher than generally anticipated and the proportion of viable viruses produced extremely small, or replication strategies cannot be optimized to maximize viral growth rate. Mechanistic models linking mutation frequency to replication mechanisms coupled with data generated through new deep-sequencing technologies could play an important role in improving the estimates of viral mutation rate.

1. Introduction

Life-history theory is a cornerstone of evolutionary biology. It mainly relies on predicting how life histories may evolve to maximize individual life-time reproductive success or intrinsic growth rate (Stearns 1992). Most studies focus on the life-history theory of multi-cellular organisms. However, the reproductive biology of microbial organisms is in some respects much simpler and more amenable to detailed understanding. In addition, some biological characteristics of viruses make them particularly well suited to the application of optimality theory: their enormous population size, high mutation rate and short generation time provide a lot of variation in life-history traits on which selection can act efficiently. Here we consider optimal replication strategies for positive-sense single-stranded RNA (ssRNA(+)) viruses within a cell.

The ssRNA(+) viruses are those in which the viral particle (virion) contains the sense strand that can be translated directly into protein. For their replication within cells, ssRNA(+) viruses must use their genetic material for at least three mutually exclusive and sequential activities. The first is translation, the production from RNA of essential viral proteins that are required for molecular replication machinery and the construction of capsids. The second is transcription, required to amplify the number of RNA strands available for further translation and/or subsequent encapsidation activities. The third is encapsidation whereby genomes are packaged into virus capsids, producing completed virions that exclude the genome from the other two processes.

Viral replication can only occur within a cell. When a single ssRNA(+) virus enters a host cell, it first undertakes translation through the use of the cellular machinery, which produces structural proteins as well as RNA-dependent RNA polymerase and any other viral proteins required for transcription (we will refer to this protein complex as ‘replicase’). If the virus is to increase its rate of protein production, it will need to generate more positive strands for translation, and these are produced through successive rounds of transcription whereby negative strands are copied from positive strands, and positive strands then copied from negative strands. As each generation of transcription occurs (negative copied from positive, and positive copied from negative), the population of positive strands increases. These positive strands can be used for (i) translation and the production of more replicase and structural proteins required for capsid assembly, (ii) further transcription and production of negative strands, or (iii) encapsidation and production of virions. At some point, the extent of viral replication within the cell causes the release of virions that go on to infect additional cells.

The optimal balance of these activities will depend on how virus fitness is defined. At least in their early stages, viral infections are characterized by a rapidly increasing viral load within an individual. It is therefore a reasonable assumption that viral fitness will be increased by rapid production of daughter virions, and that the virus should maximize its Malthusian fitness (Fisher 1930), i.e. the exponential growth rate virion production within the cell (Krakauer & Komarova 2003). Regoes et al. (2005) have addressed the question of the optimal replication strategy by examining with a simple analytical model the balance of transcription and translation that leads to maximizing the growth rate of the population of positive strands. However, their study does not consider the effects of encapsidation or mutation on this optimal strategy (figure 1). Encapsidation is required for the production of virions but it prevents further replication from the encapsidated positive strands; thus, it may play an important role in defining the optimal replication strategy.

The optimal replication strategy may also be influenced by the mutation rate, which is variously cited to be between 10−5 and 2 × 10−3 mutations per nucleotide per replication event (mut/nt/rep) among ssRNA(+) viruses (Holland et al. 1982; Drake 1993; Duffy et al. 2008). The number of mutations (many of which are deleterious) will increase through successive generations of transcription, and this may favour a replication strategy that increases the number of genomes produced from early generation transcripts (Chao et al. 2002; Drake 2007; Duffy et al. 2008). Because the mutation load accumulates with the number of transcriptional generations, the inclusion of mutation dynamics in an exact model of viral replication requires an estimate of the contribution of each transcriptional generation to the positive strand population, and assumptions about how mutation load is related to viral viability. Here, we present alternative models for viral replication within cells that include encapsidation and that incorporate the demographic contribution from different generations so that mutation dynamics can be studied.

Our analyses are based on processes typical of Picornaviridae and Potyviridae, two groups of ssRNA(+) viruses that have their genome translated into a single self-cleavable polyprotein. The Picornaviridae comprise many important animal and human viruses (e.g. Poliovirus, Hepatitis A virus, Foot-and-mouth disease virus) and the Potyviridae include many viruses infecting staple or commercial crops (e.g. Potato virus Y, Plum pox virus, Yam mosaic virus). More specifically, we chose parameters applying to Poliovirus whose molecular biology is the best understood. We address two main questions: (i) How is the optimal replication strategy affected by encapsidation and by the presence of deleterious mutation? (ii) What is the expected number of mutations in virus genomes that results from these optimal replication strategies? Our models were also used to address a series of secondary questions that include the changes through time in the ratio of positive to negative strands (or replicase) and the proportion of viable virions for different mutation rates.

2. Material and Methods

(a) Virus intracellular growth

To model virus replication, we draw heavily on the notation, parameter values and processes described by Regoes et al. (2005). All parameters used in the models described below are summarized in table 1. The within-cell treatment of positive strands assumes no complementation and is represented by the following sequence of events: (i) translation into rL self-cleavable polyproteins, each providing all the viral proteins, including one replicase and one capsid protomer (a full capsid comprises nc protomers, so each positive strand generates rL/nc full capsids), (ii) encapsidation of a proportion rL/nc of the positive strands, and (iii) transcription of the remaining free genomes into rN negative strands. As soon as a negative strand is completed, it is itself subject to transcription, which generates rP positive strands, and the process starts again. A given replication strategy is defined here by a set of values for the triplet (rN, rP, rL).

More specifically, during the first phase (translation), the cellular machinery is assumed to be non-limiting: the first ribosome moves from the 5′- to the 3′-end of the positive-sense RNA, followed by rL other evenly spaced ribosomes (with a limit of nL ribosomes fitting simultaneously on the same RNA strand). The time required by a ribosome to move along a whole positive strand and produce a single polyprotein is τL; thus, translation of the first polyprotein from a given positive strand is always completed after a delay δL = τL, and each of the subsequent rL − 1 polyproteins is fully translated after an additional delay εL = τL/nL. As a result, the time required for translation of the rL polyproteins is . The second phase (encapsidation) is assumed to happen instantaneously after the rL polyproteins have been translated from the positive strand, and precludes any further replication of the encapsidated positive strands (adding a modest delay prior to encapsidation should not change the relative fitness of different replication strategies). During the third phase (transcription), the maximum number of replicases per template strand is nx and the time taken to produce a single RNA strand is τx; thus, the first negative strand is fully transcribed from a given positive strand after a delay δN = TL + τx, and transcription of each subsequent negative strand is completed after an additional delay εN. Similarly, the first positive strand is fully transcribed from a given negative strand after a delay δP, and transcription of each subsequent positive strand is completed after an additional delay εP (figure 2). When viral replicase is not limiting, δP = τx and εN = εP = ε = τx/nx. There are three different ways (non-mutually exclusive) in which replicase can be a limiting factor and thus delay replication: (i) when rL<nx, a suboptimal number of replicases is fitted simultaneously on each positive strand template, which increases the delay between the synthesis of successive negative strands from εN = τx/nx to εN = τx/rL, (ii) when rL < nx·rN, a suboptimal number of replicases is fitted simultaneously on each negative strand template, which increases the delay between the synthesis of successive positive strands from εP = τx / nx to εP = τx·rN / rL, (iii) when rL < rN, there is not even one replicase available per negative strand template, which increases the average delay before synthesis of the first positive strand from δP = τx to

Transcriptional events involved in positive-sense single-stranded RNA virus replication. A given positive strand (in black) is first translated into proteins, and then used as a template for the transcription of rN negative strands (in grey); the first negative strand appears after a delay δN, and the time between successive negative strands is εN. Each negative strand is in turn immediately used as a template for the transcription of rP new positive strands; the first positive strand appears after a delay δP and the time between successive positive strands is εP. Here, R0 = rNrP = 8.

(b) Aggregated individual-based model of virion growth

As can be seen in figure 2, the synthesis of several strands belonging to a given generation can be completed at the same time. This property enabled the development of an aggregated individual-based model (A-IBM), which efficiently exploits the fact that when εN = εP the distribution of times when positive strands are synthesized can be computed directly (see electronic supplementary material, appendix 2). When εN ≠ εP, we used the average interval as the mean time between two consecutive strands generated from the same template strand. The A-IBM consists of three steps: (i) computing, for each generation, the distribution of times when positive strands are synthesized, (ii) encapsidating a constant proportion (rL/nc) of each generation, and (iii) summing over all generations in order to get the cumulative number of virions generated up to time t. The amount of replicase was computed directly from the distribution of the intervals between synthesis of each positive strand and synthesis of the corresponding polyproteins (and thus, replicase). As before, parameter values were taken from the literature as described by Regoes et al. (2005) (time to translate one polyprotein, τL = 6.25 min; time to transcribe one RNA strand, τx = 1.2 min; maximum number of ribosomes per RNA strand, nL = 30; maximum number of replicases per RNA strand, nx = 6.5). Viral capsids are highly symmetric in their construction, and their assembly requires multiple copies of structural proteins; for Poliovirus, the full capsid comprises nc = 60 protomers (Minor et al. 1986).

In order to compute the number of viable virions through time, the proportion of virions with a genome affected by at least one lethal mutation must be estimated. A full genome comprises S nucleotides, each of which is synthesized with a mutation rate μ; a proportion f0 of these mutations are assumed to be lethal. Thus, the random variable representing the number of lethal mutations incurred by a given genome at the gth generation of positive strands, Xg, can be modelled by a Poisson distribution with parameter λg = 2gSμf0 (the factor 2 comes from the two replication events occurring between successive generations of positive strands). The probability that a genome is viable corresponds to the null class of this distribution: . The effect of mutation was included in the A-IBM as an additional step where the total number of virions produced at generation g was multiplied by before merging the distributions corresponding to the different generations. The complete sequence of Poliovirus genome being about 7.5 kb long (Kitamura et al. 1981; Racaniello & Baltimore 1981), we took S = 7500 nucleotides. Based on previous empirical work on other RNA viruses, f0 was taken to be 0.4 (Sanjuán et al. 2004; Carrasco et al. 2007), while the other mutations were assumed to be neutral. Mutation rates are notoriously difficult to estimate for a variety of reasons (e.g. box 2 in Duffy et al. 2008); thus, in our analyses, the mutation rate characterizing Poliovirus replicase was included as a variable ranging from 10−6 to 1.1 × 10−3 mut/nt/rep with a default value of 4.5 × 10−4 mut/nt/rep, measured after just two rounds of replication with the Poliovirus polymerase (Rodriguez-Wells et al. 2001). The asymptotic growth rate of the A-IBM, estimated based on a log-linear regression, was used as a benchmark for further estimations of virion growth with an analytical model that ensures shorter computing times and exact asymptotic growth rates.

(c) Analytical model for virion growth rate

To build the analytical model, we define the total progeny number (R0) of a precursor positive strand as the number of positive strands synthesized from negative strands transcribed directly from the precursor strand and that are available for future transcription. The distribution of the times at which individual progeny are synthesized defines the generation interval distribution (see figure 2, for an example with R0 = 8) corresponding to a specific replication strategy (as defined by a set of values for rN, rP and rL). There is a mathematical relationship between R0, the growth rate r and the generation interval distribution (Wallinga & Lipsitch 2007). For the population of positive strands, any replication strategy defines R0 and the generation interval distribution, which can be represented by a histogram with equal category width ε (time between two consecutive strands in the progeny). As a result, r can be derived numerically through rearranging equation (3.6) in Wallinga & Lipsitch (2007) as: , with yi being the proportion of the progeny of one positive strand within each of n histogram categories of width ε, and a0 being the first bound of the histogram starting at ε/2 before the first progeny strand. When encapsidation is not taken into account, R0 = rNrP; with encapsidation, R0 = rNrP(1 − rL/nc). This relationship between r and R0 enabled efficient identification of the replication strategy corresponding to the optimal virion growth rate in the absence of mutation.

In the presence of mutation, the instantaneous growth rate of viable virions, rv, is negatively affected by a lethal mutation rate m. Thus, the growth rate of viable virions is rv = r − m ≈ r − λ/Tg, where λ = 2Sμf0 is the expected number of lethal mutations incurred by a progeny strand in one generation, and Tg is the mean inter-generation interval. The growth rate in the presence of mutation, rv, can either be approximated from our exact analytical model for r, in which case
2.1
or from the classical approximation r ≈ ln(R0)/Tg, in which case

2.2

In both approximations, the rescaling factor is known to be underestimated by Tg (Wallinga & Lipsitch 2007). Thus, the first approximation overestimates rv and the second one is an underestimation: we used both approximations in combination as a way to bracket rv, the growth rate of the viable virions. Note that in the absence of mutation (λ = 0), equation (2.1) gives the exact growth rate, while equation (2.2) corresponds to a classical approximation (Begon et al. 1990; Case 1999; Regoes et al. 2005).

(d) Optimization procedure

In order to find the optimal set of parameter values (for rN, rP, rL) that maximizes the Malthusian fitness (i.e. population growth rate), we used the simple optimization procedure consisting of (i) filling a cube with growth rates computed for each combination of the integer values of the three parameters (rN, rP and rL) and (ii) finding the coordinates of the maximum growth rate. After a few initial heuristic searches, rN was varied between 1 and 130, rP between 1 and 100 and rL between 1 and 30. We checked that the growth rate varied smoothly over the parameter space and that no other parameter than rP (see below) was at its optimum on the boundary of the explored parameter space.

3. Results

(a) Optimal growth rate for positive strands and virions in the absence of mutation

The analytical models enabled defining the set of parameters corresponding to the optimal replication strategy (figure 3). The maximum growth rate of positive strands (r = 0.317) corresponds to rN = 2, rP = 100 and rL = 13 (black curves). For virions, the maximum growth rate (r = 0.3) is obtained for rN = 2, rP = 100 and rL = 12 (hereafter, these values define the default replication strategy). Thus, encapsidation of rL/nc = 20 per cent of the progeny genomes only reduces the growth rate by 5.4 per cent and has almost no effect on the optimal set of parameters (dark grey curves).

Optimal growth rate conditional on the value of each parameter defining the replication strategy. (a) rP, (b) rN, and (c) rL. Growth rates of positive strands (black), virions (dark grey) and viable virions for μ = 4.5 × 10−4 mut/nt/rep (light grey for both the overestimation and underestimation) are obtained with the analytical model. Dashed lines correspond to the coordinates of the optimum, except for (a) where they correspond to 99% of the optimal growth rate. The number of negative strands, positive strands and polyproteins synthesized from one template strand are represented by rN, rP and rL, respectively.

In both cases, the optimal value for rP is at the edge of the explored parameter space, but any increase in rP leads to a very small increase in r (data not shown), so rP has in fact no finite optimal value. However, the growth rates of positive strands and virions increase asymptotically with rP (e.g. r increases by less than 10−8 when rP increases from 500 to 1000), and they reach 99 per cent of their maximum value (taken at rP = 1000) for rP = 35 and rP = 30, respectively (figure 3a, dashed lines). Thus, the optimal set of parameter values corresponds to a replication strategy in which each positive strand is transcribed into only two negative strands, each negative strand is then transcribed into as many positive strands as possible and translation is adjusted accordingly: more polyprotein synthesis (higher rL) requires too much time, while less polyprotein synthesis depletes the pool of replicase, which hinders replication.

(b) Optimal growth rate in the presence of mutation

Integrating mutation (μ = 4.5 × 10−4 mut/nt/rep) into the analytical model significantly reduces the optimal growth rate but has little impact on the optimal set of parameters, except that rN reaches higher values for both approximations (figure 3b). With increasing mutation rates, the optimal strategy shifts towards synthesizing more negative strands per positive strands and, using equation (2.1), more polyprotein (table 2). Thus, more strands of the first generation can be encapsidated, which is favoured because most of the virions from the next generations are non-viable. For the highest mutation rates, this shift leads to an optimal replication strategy that is in effect similar to a ‘stamping-machine’ replication mode (Chao et al. 2002; Duffy et al. 2008) with a very small growth rate and all the progeny genomes coming from only two transcriptional steps.

Effect of different mutation rates (mut/nt/rep) on the parameters corresponding to the maximum growth rate rv of the viable virions (bracketed by equations (2.1) and (2.2) that provide overestimated and underestimated values of rv, respectively). The number of negative strands, positive strands and polyproteins synthesized from one template strand are represented by rN, rP and rL, respectively.

(c) Distribution of mutations in the population of virions

(i) Changes in the proportion of viable virions

Virion yield increases exponentially with the number of generations (figure 4a, dashed line), but the genetic quality of the encapsidated genomes decreases exponentially at the same time. This trade-off affects the growth rate of the population of viable virions (table 2, figure 3). It also leads to an exponential decrease in the proportion of viable virions with increasing generation number, at a rate that depends on the assumed mutation rate, as shown using the A-IBM. For μ = 10−5 mut/nt/rep, the viability of most genomes is preserved, while for μ = 10−3 mut/nt/rep, the few viable virions are overwhelmed by a rapidly increasing number of non-viable virions; our default mutation rate (μ = 4.5 × 10−4 mut/nt/rep) corresponds to a decline in the proportion of viable virions from 1 to as little as 3 × 10−6 within less than five generations (figure 4a).

Effect of the exponential replication in the presence of mutation with the parameter values that maximize virion growth rate (rN = 2, rP = 100 and rL = 12). (a) Changes through successive generations in the total number of virions (dashed curve, left axis) and the proportion of viable virions for four different mutation rates μ = 10−5, 10−4, 4.5 × 10−4 and 10−3 mut/nt/rep from top to bottom (solid curves, right axis). (b) Frequency distribution of the number of mutations per encapsidated genome for four different mutation rates, at the time when 76 000 positive strands have been produced. For this 7.5 kb long genome, the average number of mutations per genome corresponds to a mutation frequency that is 7.7 higher than the mutation rate.

(ii) Frequency of mutants in the viral yield from a single cell

The peak number of positive strands within one Poliovirus-infected cell has been estimated to be approximately 76 000 (Novak & Kirkegaard 1991). By the time this value is reached in the A-IBM, each of the 15 320 encapsidated genomes has on average 26 mutations for μ = 4.5 × 10−4 mut/nt/rep (0.11% of them are viable), while mutation rates of 10−5, 10−4 and 10−3 mut/nt/rep result in an average of 0.57, 5.8 and 59 mutations per genome, respectively (figure 4b). However, a high mutation rate still produces several hundreds of viable virions, as illustrated by the A-IBM (figure 5a).

Predictions from the A-IBM, with the parameter values that maximize virion growth rate (rN = 2, rP = 100 and rL = 12). (a) Growth curves of the total number of positive strands (black), negative strands (dashed), virions (dark grey), viable virions for μ = 4.5 × 10−4 mut/nt/rep (light grey) and replicase (dotted). The straight lines correspond to the asymptotic growth curves obtained with the analytical models. (b) Damped oscillations through generations in the ratio of the number of free positive strands to the number of negative strands (black, left axis) or replicase (grey, right axis). Thin lines correspond to the asymptotic ratio.

(d) Ratio of viral molecules

(i) Ratio of free positive strands to negative strands

The A-IBM shows that the initial steps of viral replication should be dominated by transcriptional generations that overlap little because of the sequential nature of the process (figure 5a), which leads to wide oscillations in the ratio of positive to negative strands (figure 5b, black curve). For the default replication strategy, this ratio is expected to vary between 1 and 40 and to stabilize around 9.6 after some time.

(ii) Ratio of free positive strands to virions

Under the assumption that all the available capsid protein is used immediately, we could derive a straightforward expression for the ratio of positive strands to virions. For one synthesized positive strand, 1 − rL/nc remains free and rL/nc are encapsidated. Thus, the ratio of free positive strands to virions is constant and equal to nc/rL −1 = 4 (figure 5a).

(iii) Ratio of free positive strands to replicase

If we used a similar calculation, the ratio of free positive strands to replicase should be: (nc − rL)/(ncrL) = 6.7 × 10−2. However, this analytical result corresponds to the critical time when replicase may be limiting; thus, it is expected to differ from the average ratio over the whole replication cycle. The A-IBM was used to track the amount of replicase (figure 5a, dotted curve) from which we computed the ratio of free positive strands to replicase throughout generations. For the default replication strategy, the computed ratio varies between 0 and 3.5 and stabilizes around 0.59 after some time (figure 5b, grey curve). After synthesis of 12 polyproteins, each positive strand is used immediately as a template for producing two negative strands; hence the time lag and scale difference between the two curves in figure 5b.

4. Discussion

We have developed analytical and computational models of within-cell replication during the phase of exponential growth of positive-sense single-stranded RNA viruses. These new models advance our understanding of virus replication strategies in two ways: first, they explicitly include the encapsidation process; and second, because they enable the estimation of the demographic contribution of different transcriptional generations to the overall viral yield, it becomes straightforward to examine how mutation accumulation is likely to affect the number of viable virions produced from a single infected cell.

Our analytical model is in agreement with previous findings (Regoes et al. 2005) that, in the absence of encapsidation and mutation, the optimum balance of translation and transcription consists of translating each positive strand into 13 polyproteins, and then transcribing each positive strand into two negative strands. In contrast, our model predicts no upper limit to the number of positive strands transcribed from each negative strand, a finding which differs qualitatively from Regoes et al. (2005), who do find an optimal value for rP because they approximate the generation interval distribution by its mean value (resulting in a growth rate expressed as a ratio whose numerator increases logarithmically and whose denominator increases linearly with rP). However, both results are quantitatively similar since their optimal value of rP (36.6) corresponds to more than 99 per cent of the exact asymptotic growth rate. Including the encapsidation process results in a decrease of approximately 5 per cent in the growth rate of virions. Interestingly, when the capsid comprises numerous monomers (e.g. nc ∼ 2000 for the Potyviridae), the relationship R0 = rNrP(1 − rL/nc) implies that the impact of encapsidation becomes negligible on virion growth rate. In addition, the polyprotein strategy is one way, among others, to limit the rate of virus encapsidation, through producing one protomer for one replicase. The evolution of this combination of traits may have been driven by a selection pressure for high intracellular growth rates.

Replication leads to an exponentially increasing number of positive strands generated at each transcriptional generation. The addition of mutation to this process results in an exponential decrease in the proportion of genomes that are likely to be viable (figure 4a). When the mutation rate is high enough, the optimal set of parameters shifts towards a replication strategy with greater demographic contributions from the earlier generations (table 2), because the accumulated mutation load is lower. Surprisingly, our results suggest that the optimal replication strategy is broadly robust to the inclusion of mutation. Indeed, a mutation rate of 4.5 × 10−4 mut/nt/rep reduces the growth rate of viable virions by 41–60% compared with no mutation (table 2), and the proportion of viable virions to 0.11 per cent at the time when 76 000 positive strands have been produced. However, maintaining an optimal growth rate of viable virions despite increasing mutation rates requires only a modest increase in the number of negative strands produced per positive strand and in the number of polyproteins synthesized prior to the onset of transcription (table 2). Indeed, most studies of Picornavirus infections suggest that positive-sense viral RNA accumulates exponentially, at least in the early stages of cell infection (Novak & Kirkegaard 1991, 1994; Bolten et al. 1998; Li et al. 2009). Although we assumed that mutations are either lethal or neutral, the impact of slightly deleterious mutations might be considered to be cancelled by the impact of slightly beneficial mutations. More specific simulations of the effect of the fitness landscape on the observed mutation load have been conducted recently (Sardanyés et al. 2009).

Within a cell, the fittest virus might be considered as the one with the highest growth rate (as we assume here) or with the highest viral yield at a given time (which is equivalent, assuming a quick convergence towards the average growth rate). However, virus transmission between cells and at higher levels might depend on receptors that are present in limited numbers and can be blocked by non-viable virions; in such circumstances, the proportion of viable virions would be a key parameter. This would place within- and between-cell selection in conflict because the proportion and growth rate of viable virions are antagonistic for two reasons. First, there is a direct trade-off between fidelity and the polymerization rate of viral replicase (Furió et al. 2005, 2007). Second, our model has unveiled an evolutionary trade-off between the proportion and growth rate of viable virions: the strategy that maximizes the growth rate results in a minute proportion of viable viruses (figure 4a); increasing this proportion requires a move away from the optimum, towards a stamping-machine strategy, which reduces the number of transcriptional generations and thus the within-cell growth rate. In addition to this surprising role of mutation in the trade-off between the transmission and intracellular growth rate of viable virions, higher viral growth can lead to higher virulence, which is also assumed to trade-off with transmission in the classical models of virulence evolution (Coombs et al. 2003; Gilchrist et al. 2004; Alizon et al. 2009). Because of these three trade-offs, the resulting strategy might strike a compromise between the proportion and growth rate of viable virions (i.e. between quality and quantity of the progeny), or switch from a linear to an exponential replication strategy at some point in the cell-infection cycle. Aphid-transmitted plant viruses of the Potyvirus genus may be especially affected by this trade-off, because stylet-borne viruses initiate infections with so few virions (Moury et al. 2007) that the proportion of viable genomes might be crucial for transmission. Thus, we may expect such viruses to have evolved mechanisms that reduce the proportion of lethal genomes, while plant viruses transmitted after ingestion of large numbers of virions and circulation through their vector's body may tolerate a higher proportion of lethal genomes.

The optimal combination of parameters in the absence of mutation can be intuitively explained in the following way. Translation and transcription are mutually exclusive events (Gamarnik & Andino 1998), and at least initially (and here we have assumed this to apply throughout the replication cycle) translation must precede transcription, as replicase is required for transcription. Thus, delaying transcription will extend the virus generation time and reduce the growth rate, and therefore translation must be kept to the minimum required to provide the structural and non-structural proteins needed for transcription and encapsidation. A similar increase in R0 can be obtained through a given increase in the number of genomes transcribed from either positive or negative strands. However, because the same amount of replicase (translated from the positive strands) has to be used first for transcription from these positive strands and then for transcription from the resulting negative strands, the limiting factors do not act in a symmetrical way on both strands: an increase in rP results in a proportional increase in the total amount of available replicase, but any increase in rN implies a higher demand for replicase, inducing either more translation or a shortage in replicase. As both of these outcomes extend the virus generation time, the best way for the virus to increase R0 is through an increase in rP. In addition, negative strands are not required for anything other than positive strand synthesis, so this process can continue indefinitely (although the benefits of doing so become rapidly insignificant). Note that this explanation of the asymmetric replication does not involve encapsidation.

Our model predicts the time-averaged ratio of free positive to negative strands to be about 10, but suggests this ratio could be expected to fluctuate considerably in the early course of cell infection. In the model, these fluctuations arise from the discrete, deterministic and initially synchronized nature of the replication cycle. In reality, we would expect asynchronies induced by stochastic effects to damp out the oscillations much more quickly than shown in figure 5. However, it is important to note that contrary to what has been previously assumed (Regoes et al. 2005), the ratio of positive to negative strands is not expected to equate to the ratio rP/rN because the average excess in positive strands takes into account the time spent at different ratios over the whole replication cycle. For Poliovirus, a large excess (30–100) of positive strands has been measured in several studies (Andino et al. 1990; Novak & Kirkegaard 1991; Bolten et al. 1998; Paul 2002). Similar results have been obtained for other ssRNA(+) viruses, and there are indications of mechanisms that regulate the asymmetric production of positive and negative strands (Belsham 2005).

The ratio of free positive strands to replicase is predicted to be of the order of one-half, with considerable variability possible. However, the absence of any representation in the model of molecular degradation over time (through degradation either by the host proteins or by the effect of mutations) leads us to regard these ratio predictions with caution. More generally, we focused on the period of the replication dynamics during which viral genotypes actively compete, i.e. the exponential phase. Thus, we did not model the processes acting on the cell-infection cycle only at the beginning (when the virus gradually hijacks the cell machinery for its own replication) or at the end (close to the cell-carrying capacity, when nucleotides or amino acids or other components of the cell machinery can become limiting for virus replication).

Considerable uncertainty exists surrounding the rate of RNA virus mutation. Values reported can vary over three orders of magnitude (Holland et al. 1982; Parvin et al. 1986; Drake 1993; Duffy et al. 2008), and yet mutation rate is an important parameter fundamental to many questions about the evolutionary genetics of viral populations. One reason for the uncertainty in its measurement per genome replication is that while viral mutation frequency can be estimated with some precision, the number of generations of replication over which these mutants are generated is much harder to quantify. Indeed, even the definition of genome replication requires some care; it could be regarded as per transcription, or from positive strand to positive strand, or even from virion to virion. One possible future solution is to fit mutation frequency data directly to models of the sort developed here. Our increasing ability to manipulate and infect single cells, and experimentally modify mutation rates, combined with new high-throughput sequencing technologies, are likely to generate new opportunities to parametrize models of the population genetic dynamics within cells.

Acknowledgements

We thank Guillaume Martin for stimulating discussions, especially on the mutation model, and for helpful comments on the manuscript together with Sylvie Dallot, Serafín Gutiérrez, Gérard Labonne, David Pleydell and Virginie Ravigné.

Funding. This research has been funded by ANR-BBSRC SysBio (project EpiEvol).

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